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(g) (a −m ) −n = a mn (h) (ab) −n = a −n b −n 3.5 Variations of the PMI 87 EXERCISE 3.5.6 Suppose a is a nonzero real number and n is a nonnegative integer. Prove the following. 14 (a) (−a) 2n = a 2n (b) (−a) 2n+1 =−a 2n+1 EXERCISE 3.5.7 Prove the following for n ≥ 1. (a) If 0 ≤ a 1 (c) If ab2n 16 (ii) a 2n+1
88 Chapter 3 Sets and Their Properties If x �= 1, we may divide both sides through by 1 − x to have 1 + x + x 2 + x 3 + x 4 = 1 − x5 1 − x This algebraic observation suggests a general formula for the sum of powers of a real number x �= 1. If n ≥ 0, then 1 + x + x 2 + x 3 +···+x n = n� x k = k=0 Prove Eq. (3.51) with an induction argument on n ≥ 0. 1 − xn+1 1 − x (3.51) EXERCISE 3.5.11 Prove the following factorization formula for nonzero real numbers a and b and nonnegative integer n. 18 a n+1 − b n+1 = (a − b)(a n + a n−1 b + a n−2 b 2 +···+a 2 b n−2 + ab n−1 + b n ) (3.52) Another reason Theorem 3.5.1 is useful is that a result might not be true for all positive integers, but only eventually true, that is, true for n ≥ j for some positive integer j. Just like we use � n k=1 ak to denote the sum of the ak, we can use the notation � n k=1 ak to denote the product of the terms. In other words, � n k=1 ak = a1 · a2 · a3 ···an EXERCISE 3.5.12 For all n ≥ 2, �n � � 1 1 k=2 1 − k = n . EXERCISE 3.5.13 If n ≥ 4 is an integer, then 2 n 2n + 1. Proof. Let n ≥ 3 be an integer. Then n 2 = n · n ≥ 3n = 2n + n>2n + 1 EXERCISE 3.5.16 If n ≥ 5 is an integer, then n 2 < 2 n . 18 Don’t make another induction argument. Letting x = b/a in Eq. (3.51) provides a good start.
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A Transition to Abstract Mathematic
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A Transition to Abstract Mathematic
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For Topo my little mouse
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Contents Why Read This Book? xiii P
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3.10 Irrational Numbers 107 3.11 Re
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8.2 Subgroups 252 8.2.1 Subgroups D
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Why Read This Book? One of Euclid
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Preface A Transition to Abstract Ma
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Preface to the First Edition This t
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Preface to the First Edition xix ea
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Acknowledgments It takes an entire
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Notation and Assumptions 0 Suppose
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0.2 Assumptions about the Real Numb
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0.2 Assumptions about the Real Numb
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0.2.3 Other Assumptions 0.2 Assumpt
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P A R T I Foundations of Logic and
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Language and Mathematics 1 One main
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1.1 Introduction to Logic 13 Now im
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The compound statement we call “p
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1.1 Introduction to Logic 17 exampl
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1.2 If-Then Statements 19 Definitio
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1.2 If-Then Statements 21 (h) The o
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1.2 If-Then Statements 23 (g) If ei
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p : Meghan is at least 25 years old
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1.3 Universal and Existential Quant
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1.4 Negations of Statements 33 2. F
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1.4 Negations of Statements 35 want
Page 60 and 61: Solution 1.4 Negations of Statement
Page 62 and 63: Solution 1. Everyone in this class
Page 64 and 65: 1.5 How We Write Proofs 41 Our purp
Page 66 and 67: 1.5 How We Write Proofs 43 the nega
Page 68 and 69: Properties of Real Numbers 2 It’s
Page 70 and 71: 2.1 Basic Algebraic Properties of R
Page 72 and 73: 2.1.2 Properties of Multiplication
Page 74 and 75: 2.2 Ordering Properties of the Real
Page 76 and 77: (c) For all real numbers a, a2 ≥
Page 78 and 79: 2.3 Absolute Value 55 (⇐) Now sup
Page 80 and 81: 2.4 The Division Algorithm 57 mn =
Page 82 and 83: 2.5 Divisibility and Prime Numbers
Page 84 and 85: 2.5 Divisibility and Prime Numbers
Page 86 and 87: Sets and Their Properties 3 All of
Page 88 and 89: U A B Figure 3.5 Venn diagram illus
Page 90 and 91: (f) For all sets A, B, and C,ifA
Page 92 and 93: 3.2 Proving Basic Set Properties 69
Page 94 and 95: 3.3 Families of Sets 71 Notice that
Page 96 and 97: Example 3.3.2 Consider the family o
Page 98 and 99: 3.3 Families of Sets 75 x ∈ � A
Page 100 and 101: 3.3 Families of Sets 77 and ... and
Page 102 and 103: Example 3.4.2 For any positive inte
Page 104 and 105: 3.4 The Principle of Mathematical I
Page 106 and 107: Proof. To clean up the notation, we
Page 108 and 109: 3.5 Variations of the PMI 85 EXERCI
Page 112 and 113: Strong Induction 3.5 Variations of
Page 114 and 115: 3.6 Equivalence Relations 91 EXERCI
Page 116 and 117: 3.6 Equivalence Relations 93 beginn
Page 118 and 119: 3.6 Equivalence Relations 95 constr
Page 120 and 121: 3.7 Equivalence Classes and Partiti
Page 126 and 127: 3.8 Building the Rational Numbers 1
Page 128 and 129: 3.8 Building the Rational Numbers 1
Page 130 and 131: 3.10 Irrational Numbers 107 EXERCIS
Page 132 and 133: 3.10 Irrational Numbers 109 To the
Page 134 and 135: EXERCISE 3.10.2 √ 2 is irrational
Page 136 and 137: (a) {(x, y) : x ≤ y} (b) {(x, y)
Page 138 and 139: 3.11 Relations in General 115 (O3)
Page 140 and 141: 3.11 Relations in General 117 at al
Page 142 and 143: Functions 4 Second only to sets, fu
Page 144 and 145: 4.1 Definition and Examples 121 pro
Page 146 and 147: Solution We show that T satisfies p
Page 148 and 149: 4.2 One-to-one and Onto Functions 4
Page 150 and 151: 4.2 One-to-one and Onto Functions 1
Page 152 and 153: A 1 x A B f (A 1 ) Figure 4.4 The i
Page 154 and 155: 4.4 Composition and Inverse Functio
Page 156 and 157: 4.4 Composition and Inverse Functio
Page 158 and 159: 4.5 Three Helpful Theorems 135 The
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4.6 Finite Sets 137 EXERCISE 4.5.3
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4.7 Infinite Sets 139 The next exer
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4.7 Infinite Sets 141 of A. This or
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4.7 Infinite Sets 143 EXERCISE 4.7.
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EXERCISE 4.8.3 If A1,A2, and B are
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4.8 Cartesian Products and Cardinal
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4.8 Cartesian Products and Cardinal
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4.9 Combinations and Partitions 151
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4.10 The Binomial Theorem 157 EXERC
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EXERCISE 4.10.3 Determine the follo
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(c) The coefficient of ab 3 c 2 in
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P A R T I I Basic Principles of Ana
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The Real Numbers 5 Let’s look at
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5.1 The Least Upper Bound Axiom 167
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5.2 The Archimedean Property 169 EX
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5.2 The Archimedean Property 171 No
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5.3 Open and Closed Sets 173 Thus A
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5.4 Interior, Exterior, Boundary, a
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5.4 Interior, Exterior, Boundary, a
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5.5 Closure of Sets 179 The constru
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5.6 Compactness 181 EXERCISE 5.6.3
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5.6 Compactness 183 EXERCISE 5.6.13
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Sequences of Real Numbers 6.1 Seque
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6.1 Sequences Defined 187 Example 6
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EXERCISE 6.1.15 Consider the sequen
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L 1 e L L 2 e . . . . . . 1 2 3 4 N
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6.2 Convergence of Sequences 193 EX
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6.2 Convergence of Sequences 195 To
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6.3 The Nested Interval Property 19
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a1 a4 aN 1 2 aN aN 1 1 aN 1 3 a5 a3
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Example 6.4.7 The terms a0 = 1 a1 =
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Functions of a Real Variable 7 Func
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7.1 Bounded and Monotone Functions
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50 40 30 20 7 1 ε 7 7 2 ε 0 ( ( F
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7.2 Limits and Their Basic Properti
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7.2 Limits and Their Basic Properti
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7.3 More on Limits 217 values of 0
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7.4 Limits Involving Infinity 219 W
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7.4 Limits Involving Infinity 221 T
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(a) limx→a f(x) =−∞ (b) limx
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7.5 Continuity 225 If f is not cont
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1 2 3 4 5 6 Figure 7.6 Some example
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|a − x| |xa| = |x − a| |x||a| 7
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7.6 Implications of Continuity 231
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7.6 Implications of Continuity 233
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7.7 Uniform Continuity 235 we decla
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7.7 Uniform Continuity 237 Next, le
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7.7.2 Uniform Continuity and Compac
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P A R T I I I Basic Principles of A
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Groups 8 In its simplest terms, alg
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8.1 Introduction to Groups 245 are
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8.1 Introduction to Groups 247 Defi
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◦ 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 0
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Show that C × is an abelian group
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8.2 Subgroups 253 G1 and G3 are aut
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8.2 Subgroups 255 Notice how proper
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8.2 Subgroups 257 Example 8.2.15 su
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EXERCISE 8.2.23 If G is a group and
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8.3 Quotient Groups 261 [0] ={...,
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8.3 Quotient Groups 263 is standard
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Step 2: Define an operation ∗H on
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(iii) (Z + 3.8) +Z (Z + 1.2) (iv)
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then for a given f ∈ S and any a
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Thus f 2 = f ◦ f = (132)(56) ◦
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2 3 1 3 1 Figure 8.1 Rigid square u
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EXERCISE 8.4.12 Complete Table 8.64
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8.5 Normal Subgroups 277 Theorem 8.
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8.5 Normal Subgroups 279 point to b
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8.6 Group Morphisms 281 EXERCISE 8.
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8.6 Group Morphisms 283 Notice that
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Ker � e x 2 x maps to x Ker � G
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Rings 9 We can create algebraic str
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9.1 Rings and Fields 289 (R9) Multi
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9.1 Rings and Fields 291 we define
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9.2 Subrings 293 In addition to pro
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9.2 Subrings 295 p1 = q1 = 3. Verif
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9.3 Ring Properties 297 Theorem 9.3
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9.3 Ring Properties 299 EXERCISE 9.
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9.4 Ring Extensions 301 in the inte
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9.4 Ring Extensions 303 Example 9.4
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9.4 Ring Extensions 305 We insist t
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9.5 Ideals 307 An ideal, say a left
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9.6 Generated Ideals 309 Proof. Sup
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EXERCISE 9.6.6 In the integers, wha
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9.7 Prime and Maximal Ideals 313 Ev
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9.8 Integral Domains 315 EXERCISE 9
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9.8 Integral Domains 317 Corollary
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9.9 Unique Factorization Domains 31
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9.10 Principal Ideal Domains 321 al
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9.10 Principal Ideal Domains 323 So
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9.11 Euclidean Domains 325 next exe
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9.11 Euclidean Domains 327 Exercise
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Proof. Let f and g be nonzero polyn
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9.12 Polynomials over a Field 331 W
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9.13 Polynomials over the Integers
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9.14 Ring Morphisms 335 Example 9.1
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9.14 Ring Morphisms 337 EXERCISE 9.
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9.15 Quotient Rings 339 this, howev
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9.15 Quotient Rings 341 Because the
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9.15 Quotient Rings 343 However, th
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Index =,51 ɛ (epsilon), 167-168 ɛ
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Index 347 Cosets, 263, 264, 265 Lag
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Index 349 Gauchy sequences, 202-206
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Index 351 identity, 124 relations v
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Index 353 Quaternions, 248, 253, 25
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Index 355 Tower of Hanoi, 84 Transc
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